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Functional *N*-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories ()

The

*N*-representability conditions on the reduced second-order reduced density matrix (2-RDM), impose restrictions not only in the context of reduced density matrix theory (RDMT), but also on functionals advanced in one-matrix theory such as natural orbital functional theory (NOFT), and on functionals depending on the one-electron density such as those of density functional theory (DFT). We review some aspects of the applications of these*N*-representability conditions in these theories and present some conclusions.Share and Cite:

E. Ludeña, F. Torres and C. Costa, "Functional

*N*-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories,"*Journal of Modern Physics*, Vol. 4 No. 3A, 2013, pp. 391-400. doi: 10.4236/jmp.2013.43A055.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | D. A. Mazziotti, “Two-Electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics,” Chemical Review, Vol. 112, No. 1, 2012, pp. 244-262. doi:10.1021/cr2000493 |

[2] | M. Nakata, M. Fukuda and K. Fujisawa, “Variational Approach to Electronic Structure Calculations on Second-Order Reduced Density Matrices and the N-Representability Problem,” Lecture Note Series, Vol. 9, 2012, pp. 1-32. |

[3] | P. A. M. Dirac, “Quantum Mechanics of Many Electron Systems,” Proceedings of the Royal Society A, Vol. 123, No. 792, 1929, pp. 714-735 |

[4] | P.-O. Lowdin, “Quantum Theory of Many-Particle Systems I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction,” Physical Review, Vol. 97, No. 6, 1955, pp. 1474-1489. doi:10.1103/PhysRev.97.1474 |

[5] | R. McWeeny, “Some Recent Advances en Density Matrix Theory,” Reviews of Modern Physics, Vol. 32, No. 2, 1960, pp. 335-369. doi:10.1103/RevModPhys.32.335 |

[6] | P. A. M. Dirac, “Note on Exchange Phenomena in the Thomas Atom,” Proceedings of Cambridge Philosophy Society, Vol. 26, No. 3, 1930, pp. 376-385. doi:10.1017/S0305004100016108 |

[7] | P. A. M. Dirac, “Note on the Interpretation of the Density Matrix in the Many-Electron Problem,” Proceedings of Cambridge Philosophical Society, Vol. 27, No. 2, 1931, pp. 240-243. doi:10.1017/S0305004100010343 |

[8] | K. Husimi, “Some Formal Properties of the Density Matrix,” Proceedings of Physical Mathematical Society of Japan, Vol. 22, 1940, pp. 264-314. |

[9] | J. E. Mayer, “Electron Correlation,” Physical Review, Vol. 100, No. 6, 1955, pp. 1579-1586. |

[10] | R. H. Tredgold, “Density Matrix and the Many-Body Problem,” Physical Review, Vol. 105, No. 5, 1957, pp. 1421-1423. doi:10.1103/PhysRev.105.1421 |

[11] | A. J. Coleman, “Structure of Fermion Density Matrices,” Reviews of Modern Physics, Vol. 35, No. 3, 1963, pp. 668-687. doi:10.1103/RevModPhys.35.668 |

[12] | A. J. Coleman, “The Convex Structure of Electrons,” International Journal of Quantum Chemistry, Vol. 11, No. 6, 1977, pp. 907-916. doi:10.1002/qua.560110604 |

[13] | A. J. Coleman, “Reduced Density Operators and the N-Particle Problem,” International Journal of Quantum Chemistry, Vol. 13, No. 1, 1978, pp. 67-82. doi:10.1002/qua.560130106 |

[14] | C. Garrod and J. K. Percus, “Reduction of the N-Particle Variational Problem,” Journal of Mathematical Physics, Vol. 5, No. 12, 1964, pp. 1756-1776. doi:10.1063/1.1704098 |

[15] | E. R. Davidson, “Reduced Density Matrices in Quantum Chemistry,” Academic Press, London, 1976. |

[16] | J. Cioslowski, “Many-Electron Densities and Reduced Density Matrices,” Kluwer, New York, 2000. |

[17] | A. J. Coleman and V. I. Yukalov, “Reduced Density Matrices: Coulson’s Challenge,” Springer-Verlag, New York, 2000. doi:10.1007/978-3-642-58304-9 |

[18] | L. Cohen and C. Frishberg, “Hartree-Fock Density Matrix Equation,” Physical Review A, Vol. 13, No. 3, 1976, pp. 4234-4238. doi:10.1103/PhysRevA.13.927 |

[19] | H. Nakatsuji, “Equation for the Direct Determination of the Density Matrix,” Physical Review A, Vol. 14, No. 1, 1976, pp. 41-50. doi:10.1103/PhysRevA.14.41 |

[20] | D. R. Alcoba and C. Valdemoro, “Family of Modified-Contracted Schrodinger Equations,” Physical Review A, Vol. 64, No. 6, 2001, Article ID: 062105. |

[21] | J. E. Harriman, “Limitation on the Density-Equation Approach to Many-Electron Problems,” Physical Review A, Vol. 19, No. 5, 1979, pp. 1893-1895. doi:10.1103/PhysRevA.19.1893 |

[22] | W. Kutzelnigg, “Generalized K-Particle Brillouin Conditions and Their Use for the Constructlon of Correlated Electronic Wavefunctions,” Chemical Physics Letters, Vol. 64, No. 2, 1979, pp. 383-387. doi:10.1016/0009-2614(79)80537-0 |

[23] | C. Valdemoro, “Approximating the Second-Order Reduced Density Matrix in Terms of the First-Order One,” Physical Review A, Vol. 45, No. 7, 1992, pp. 4462-4467. doi:10.1103/PhysRevA.45.4462 |

[24] | F. Colmenero and C. Valdemoro, “Approximating q-Order Reduced Density Matrices in Terms of the Lower-Order Ones. II. Applications,” Physical Review A, Vol. 47, No. 2, 1993, pp. 979-987. doi:10.1103/PhysRevA.47.979 |

[25] | H. Nakatsuji and K. Yasuda, “Direct Determination of the Quantum-Mechanical Density Matrix Using the Density Equation,” Physical Review Letters, Vol. 76, No. 7, 1996, pp. 1039-1042. doi:10.1103/PhysRevLett.76.1039 |

[26] | D. A. Mazziotti, “Contracted Schrodinger Equation: Determining Quantum Energies and Two-Particle Density Matrices without Wave Functions,” Physical Review A, Vol. 57, No. 7, 1998, pp. 4219-4243. doi:10.1103/PhysRevA.57.4219 |

[27] | D. A. Mazziotti, “Approximate Solution for Electron Correlation through the Use of Schwinger Probes,” Chemical Physics Letters, Vol. 289, No. 6, 1998, pp. 419-427. doi:10.1016/S0009-2614(98)00470-9 |

[28] | D. A. Mazziotti, “Complete Reconstruction of Reduced Density Matrices,” Chemical Physics Letters, Vol. 326, No. 3-4, 2000, pp. 212-218. doi:10.1016/S0009-2614(00)00773-9 |

[29] | D. A. Mazziotti, “Pursuit of N-Representability for the Contracted Schrodinger Equation through Density-Matrix Reconstruction,” Physical Review A, Vol. 60, No. 5, 1999, pp. 3618-3626. doi:10.1103/PhysRevA.60.3618 |

[30] | W. Kutzelnigg and D. Mukherjee, “Direct Determination of the Cumulants of the Reduced Density Matrices,” Chemical Physics Letters, Vol. 317, No. 6, 2000, pp. 567-574. doi:10.1016/S0009-2614(99)01410-4 |

[31] | M. Nooijen, M. Wladyslawski and A. Hazra, “Cumulant Approach to the Direct Calculation of Reduced Density Matrices: A Critical Analysis,” Journal of Chemical Physics, Vol. 118, No. 11, 2003, pp. 4832-4848. doi:10.1063/1.1545779 |

[32] | D. R. Alcoba and C. Valdemoro, “The Correlation Contracted Schrdinger Equation: An Accurate Solution of the G-Particle-Hole Hypervirial,” International Journal of Quantum Chemistry, Vol. 109, No. 14, 2009, pp. 3178-3190. doi:10.1002/qua.21943 |

[33] | D. A. Mazziotti, “Parametrization of the Two-Electron Reduced Density Matrix for Its Direct Calculation without the Many-Electron Wave Function: Generalizations and Applications,” Physical Review A, Vol. 81, No. 6, 2010, Article ID: 062515. |

[34] | L. Vandenbergue and S. Boyd. “Semidefinite Programming,” SIAM Review, Vol. 38, No. 1, 1996, pp. 49-50. doi:10.1137/1038003 |

[35] | M. J. Todd, “Semidefinite Optimization,” Acta Numerals, Vol. 10, 2001, pp. 515-560. doi:10.1017/S0962492901000071 |

[36] | E. A. Yildirim and M. J. Todd, “Sensitivity Analysis in Linear Programming and Semidefinite Programming Using Interior-Point Methods,” Mathematical Programming, Vol. 90, No. 2, 2001, pp. 229-261. doi:10.1007/PL00011423 |

[37] | S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge University Press, Cambridge, 2009. |

[38] | S. Pironio, M. Navascues and A. Acio, “Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables,” SIAM Journal of Optimization, Vol. 20, No. 5, 2010, pp. 2157-2180. doi:10.1137/090760155 |

[39] | M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata and K. Fujisawa, “Variational Calculations of Fermion Second-Order Reduced Density Matrices by Semidefinite Programming Algorithm,” Journal of Chemical Physics, Vol. 114, No. 19, 2001, pp. 8282-8292. doi:10.1063/1.1360199 |

[40] | D. A. Mazziotti and R. M. Erdahl, “Uncertainty Relations and Reduced Density Matrices: Mapping Many-Body Quantum Mechanics onto Four Particles,” Physical Review A, Vol. 63, No. 4, 2001, Article ID: 042113. |

[41] | S. Burer and R. D. C. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming, Vol. 95, 2003, pp. 329-357. doi:10.1007/s10107-002-0352-8 |

[42] | D. A. Mazziotti, “Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming,” Physical Review Letters, Vol. 93, No. 21, 2004, Article ID: 213001. |

[43] | D. A. Mazziotti, “Exactness of Wave Functions from Two-Body Exponential Transformations in Many-Body Quantum Theory,” Physical Review A, Vol. 69, No. 1, 2004, Article ID: 012507. |

[44] | M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton, J. K. Percus, M. Yamashita and Z. Zhao, “Large-Scale Semidefinite Programs in Electronic Structure Calculation,” Mathematical Programming, Vol. 109, 2007, pp. 553-580. doi:10.1007/s10107-006-0027-y |

[45] | M. Fukuda, M. Nakata and M. Yamashita, “Semidefinite Programming: Formulations and Primal-Dual Interior-Point Methods,” Advanced Chemical Physics, Vol. 134, 2007, pp. 103-118. |

[46] | M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita and Z. Zhao, “Variational Calculation of Second-Order Reduced Density Matrices by Strong N-Representability Conditions and an Accurate Semidefinite Programming Solver,” Journal of Chemical Physics, Vol. 128, No. 16, 2008, Article ID: 164113. |

[47] | M. Yamashita, K. Fujisawa, M. Fukuda, K. Nakata and M. Nakata, “Algorithm 925: Parallel Solver for Semidefinite Programming Problem Having Sparse Schur Complement Matrix,” ACM Transactions on Mathematical Software, Vol. 39, No. 1, 2012, pp. 1-22. doi:10.1145/2382585.2382591 |

[48] | T. Baumgratz and M. B. Plenio, “Lower Bounds for Ground States of Condensed Matter Systems,” New Journal of Physics, Vol. 14, Article ID: 023027. |

[49] | D. A. Mazziotti, “Variational Minimisation of Atomic and Molecular Ground-State Energies via the Two-Particle Reduced Density Matrix,” Physical Review A, Vol. 65, No. 6, 2002Article ID: 062511. |

[50] | D. A. Mazziotti, “Variational Method for Solving the Contracted Schrodinger Equation through a Projection of the N-Particle Power Method onto the Two-Particle Space,” Journal of Chemical Physics, Vol. 116, No. 4, 2002, pp. 1239-1249. doi:10.1063/1.1430257 |

[51] | G. Guidofalvi and D. A. Mazziotti, “Boson Correlation Energies via Variational Minimization with the Two-Particle Reduced Density Matrix: Exact N-Representability Conditions for Harmonic Interactions,” Physical Review A, Vol. 69, No. 4, 2004, Article ID: 042511. |

[52] | K. Yasuda, “Uniqueness of the Solution of the Contracted Schrodinger Equation,” Physical Review A, Vol. 65, No. 5, 2002, Article ID: 052121. |

[53] | M. Nakata, M. Ehara and H. Nakatsuji, “Fundamental World of Quantum Chemistry,” Kluwer Academic Publishers, New York, 2003. |

[54] | D. A. Mazziotti, “Variational Two-Electron Reduced Density Matrix Theory for Many-Electron Atoms and Molecules: Implementation of the Spin- and Symmetry-Adapted T2 Condition through First-Order Semidefinite Programming,” Physical Review A, Vol. 72, No. 3, 2005, Article ID: 032510. |

[55] | G. Guidofalvi and D. A. Mazziotti, “Computation of Quantum Phase Transitions by Reduced-Density-Matrix Mechanics,” Physical Review A, Vol. 74, No. 1, 2006, Article ID: 012501. |

[56] | D. R. Alcoba, C. Valdemoro, L. M. Tel and E. Pérez-Romero, “Controlling the N- and S-Representability of the Second-Order Reduced Density Matrix: The Doublet-State Case,” Physical Review A, Vol. 77, No. 4, 2008, Article ID: 042508. |

[57] | Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton and J. K. Percus, “The Reduced Density Matrix Method for Electronic Structure Calculations and the Role of Three-Index Representability,” Journal of Chemical Physics, Vol. 120, No. 5, 2004, pp. 2095-2125. doi:10.1063/1.1636721 |

[58] | M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita and Z. Zhao, “Variational Calculation of Second-Order Reduced Density Matrices by Strong N-Representability Conditions and an Accurate Semidefinite Programming Solver,” Journal of Chemical Physics, Vol. 128, No. 16, 2008, Article ID: 164113. |

[59] | D. A. Mazziotti, “Structure of Fermionic Density Matrices: Complete N-Representability Conditions,” Physical Review Letters, Vol. 108, No. 26, 2012, Article ID: 263002. |

[60] | D. A. Mazziotti, “Significant Conditions for the Two-Ele- ctron Reduced Density Matrix from the Constructive Solution of N-Representability,” Physical Review A, Vol. 85, No. 6, 2012, Article ID: 062507. |

[61] | M. Levy, “Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the V-Representability Problem,” Proceedings of National Academy Science of the USA, Vol. 76, No. 12, 1979, pp. 6062-6065. doi:10.1073/pnas.76.12.6062 |

[62] | T. L. Gilbert, “Hohenberg-Kohn Theorem for Nonlocal Externai Potentiais,” Physical Review B, Vol. 12, No. 6, 1975, pp. 2111-2120. doi:10.1103/PhysRevB.12.2111 |

[63] | R. A. Donnelly and R. G. Parr, “Elementary Properties of an Energy Functional of the First Order Reduced Density Matrix,” Journal of Chemical Physics, Vol. 69, No. 10, 1978, pp. 4431-4439. doi:10.1063/1.436433 |

[64] | R. A. Donnelly, “On a Fundamental Difference between Energy Functionals Based on First and on Second Order Density Matrices,” Journal of Chemical Physics, Vol. 71, No. 7, 1979, pp. 2874-2879. doi:10.1063/1.438678 |

[65] | S. M. Valone, “Consequences of Extending 1matrix Energy Functionals from Pure State Representable to All Ensemble Representable 1 Matrices,” Journal of Chemical Physics, Vol. 73, No. 3, 1980, pp. 1344-1349. doi:10.1063/1.440249 |

[66] | T. T. Nguyen-Dang, E. V. Ludena and Y. Tal, “Variation of the Energy Functional of the Reduced 1st-Order Density Operator,” Journal of Molecular Structures (Theochem), Vol. 120, 1985, pp. 247-264. doi:10.1016/0166-1280(85)85114-9 |

[67] | E. V. Ludena and A. Sierraalta, “Necessary Conditions for the Mapping of Gamma-into ρ,” Physical Review A, Vol. 32, No. 1, 1985, pp. 19-25. doi:10.1103/PhysRevA.32.19 |

[68] | E. V. Ludena, “Density Matrices and Density Functionals,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3855-7_15 |

[69] | M. Piris, “Natural Orbital Functional Theory,” In: D. A. Mazziotti, Ed., Reduced Density-Matrix Mechanics with Applications to Many-Electron Atoms and Molecules, John Wiley and Sons, New York, 2007, p. 387. doi:10.1002/9780470106600.ch14 |

[70] | A. M. K. Müller, “Explicit Approximate Relation between Reduced Two- and One-Particle Density Matrices,” Physical Letter A, Vol. 105, No. 9, 1984, pp. 446-452. doi:10.1016/0375-9601(84)91034-X |

[71] | S. Goedecker and C. J. Umrigar, “A Natural Orbital Functional for the Many-Electron Problem,” Physical Review Letters, Vol. 81, No. 4, 1998, pp. 866-870. doi:10.1103/PhysRevLett.81.866 |

[72] | G. Csanyi and T. A. Arias, “Tensor Product Expansions for Correlation in Quantum Many-Body Systems,” Physical Review B, Vol. 61, No. 11, 2000, pp. 7348-7352. doi:10.1103/PhysRevB.61.7348 |

[73] | E. J. Baerends, “Exact Exchange-Correlation Treatment of Dissociated H2 in Density Functional Theory,” Physical Review Letters, Vol. 87, No. 13, 2001, Article ID: 133004. |

[74] | M. A. Buijse and E. J. Baerends, “An Approximate Exchange-Correlation Hole Density as a Functional of the Natural Orbitals,” Molecular Physics, Vol. 100, No. 4, 2002, pp. 401-421. doi:10.1103/PhysRevB.61.7348 |

[75] | O. Gritsenko, K. Pernal and E. J. Baerends, “An Improved Density Matrix Functional by Physically Motivated Repulsive Corrections,” Journal of Chemical Physics, Vol. 122, No. 20, 2005, Article ID: 204102. |

[76] | S. Sharma, J. K. Dowhurst, N. N. Lathiotakis and E. K. U. Gross, “Reduced Density Matrix Functional for Many-Electron Systems,” Physical Review B, Vol. 78, No. 20, 2008, Article ID: 201103. |

[77] | N. N. Lathiotakis, N. Helbig and E. K. U. Gross, “Performance of One-Body Reduced Density-Matrix Functionals for the Homogeneous Electron Gas,” Physical Review B, Vol. 75, No. 19, 2007, Article ID: 195120. |

[78] | N. N. Lathiotakis and M. A. L. Marques, “Benchmark Calculations for Reduced Density-Matrix Functional Theory,” Journal of Chemical Physics, Vol. 128, No. 18, 2008, Article ID: 184103. |

[79] | N. N. Lathiotakis, N. Helbig, A. Zacarias and E. K. U. Gross, “A Functional of the One-Body-Reduced Density Matrix Derived from the Homogeneous Electron Gas: Performance for Finite Systems,” Journal of Chemical Physics, Vol. 130, No. 6, 2009, Article ID: 064109. |

[80] | N. N. Lathiotakis, N. I. Guidopoulos and N. Helbig, “Size Consistency of Explicit Functionals of the Natural Orbitals in Reduced Density Matrix Functional Theory,” Journal of Chemical Physics, Vol. 132, No. 8, 2010, Article ID: 084105. |

[81] | R. L. Frank, E. H. Lieb, R. Seiringer and H. Siedentrop, “Müller’s Exchange-Correlation Energy in Density-Matrix-Functional Theory,” Physical Review A, Vol. 76, No. 5, 2007, Article ID: 052517. |

[82] | M. Piris, “A New Approach for the Two-Electron Cumulant in Natural Orbital Functional Theory,” International Journal of Quantum Chemistry, Vol. 106, No. 5, 2006, pp. 1093-1104. doi:10.1002/qua.20858 |

[83] | M. Piris, “Natural Orbital Functional Theory: Molecules and Polymers,” Recent Research Development of Quantum Chemistry, Vol. 4, 2004, pp. 1-26. |

[84] | M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, “Communication: The Role of the Positivity N-Representability Conditions in Natural Orbital Functional Theory,” Journal of Chemical Physics, Vol. 133, No. 11, 2010, Article ID: 111101. |

[85] | M. Piris, X. Lopez, F. Ruizpérez, J. M. Matxain and J. M. Ugalde, “A Natural Orbital Functional for Multiconfigurational States,” Journal of Chemical Physics, Vol. 134, No. 16, 2011, Article ID: 164102. |

[86] | J. M. Matxain, M. Piris, J. Uranga, X. Lopez, G. Merino and J. M. Ugalde, “The Nature of Chemical Bonds from PNOF5 Calculations,” Chemical Physics, Vol. 13, No. 9, 2012, pp. 2297-2303. doi:10.1002/cphc.201200205 |

[87] | M. Piris, “A Natural Orbital Functional Based on an Explicit Approach of the Two-Electron Cumulant,” International Journal of Quantum Chemistry, Vol. 113, No. 5, 2013, pp. 620-630. doi:10.1002/qua.24020 |

[88] | K. Pernal, “The Equivalence of the Piris Natural Orbital Functional 5 (PNOF5) and the Antisymmetrized Product of Strongly Orthogonal Geminal Theory,” Computational and Theoretical Chemistry, Vol. 1003, 2013, pp. 127-129. doi:10.1016/j.comptc.2012.08.022 |

[89] | E. V. Ludena, F. Illas and A. Ramrez-Sols, “On the N-Representability and Universality of F[ρ] in the Hohenberg-Kohn-Sham Version of Density Functional Theory,” International Journal of Modern Physics, Vol. 22, No. 25-26, 2008, pp. 4642-4654. doi:10.1142/S0217979208050395 |

[90] | P.-O. Lowdin, “Density Matrices and Density Functionals,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3855-7_3 |

[91] | R. McWeeny, “Density-Functions and Density Functionals,” Philosophical Magazine Part B, Vol. 69, No. 5, 1994, pp. 727-735. doi:10.1080/01418639408240141 |

[92] | E. V. Ludena and J. Keller, “The Importance of Pure-State N-Representability in the Derivation of Extended Kohn-Sham Equations,” Advaced Quantum Chemistry, Vol. 21, 1990, pp. 46-67. |

[93] | E. S. Kryachko and E. V. Ludena, “Formulation of N-Representable and V-Representable Density-Functional Theory,” Physical Review A, Vol. 43, No. 5, 1991, pp. 2179-2193. doi:10.1103/PhysRevA.43.2179 |

[94] | E. S. Kryachko and E. V. Ludena, “The N-Representability Problem and the Local-Scaling Version of Density Functional Theory,” Condensed Matter Theories, Vol. 7, pp. 229-241. doi:10.1007/978-1-4615-3352-8_22 |

[95] | E. V. Ludena, V. V. Karasiev, A. Artemyev and D. Gomez, “Many-Electron Densities and Reduced Density Ma- trices,” Kluwer, New York, 2000. doi:10.1007/978-1-4615-4211-7_10 |

[96] | O. Bokanowski, “New N-Representability Results Involving Symmetry and Application to the Density-Functional Theory Formalism,” Journal of Mathematical Chemistry, Vol. 26, No. 4, 1999, pp. 271-296. doi:10.1023/A:1019106516542 |

[97] | P. W. Ayers and S. Liu, “Necessary and Sufficient Conditions for the N-Representability of Density Functionals,” Physical Review A, Vol. 75, No. 2, 2007, Article ID: 022514. |

[98] | P. W. Ayers, R. Cuevas-Saavedra and D. Chakraborty, “A Variational Principle for the Electron Density Using the Exchange Hole & Its Implications for N-Representability,” Physical Letters A, Vol. 376, No. 6-7, 2012, pp. 839-844. doi:10.1016/j.physleta.2012.01.028 |

[99] | W. Kurlancheek and M. Head-Gordon, “Violations of N-Representability from Spin-Unrestricted Orbitals in M?ller-Plesset Perturbation Theory and Related Double-Hybrid Density Functional Theory,” Molecular Physics, Vol. 107, No. 8-12, 2009, pp. 1223-1232. doi:10.1080/00268970902835637 |

[100] | C. Lee, W. Yang and R. G. Parr, “Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Elcetron Density,” Physical Review B, Vol. 37, No. 2, 1988, pp. 785-789. doi:10.1103/PhysRevB.37.785 |

[101] | R. Colle and O. Salvetti, “Approximate Calculation of the Correlation Energy or the Closed Shells,” Theoretica Chimica Acta, Vol. 37, No. 4, 1975, pp. 329-334. doi:10.1007/BF01028401 |

[102] | B. Miehlich, A. Savin, H. Stoll and H. Preuss, “Results Obtained with the Correlation-Energy Density Functionals of Becke and Lee, Yang and Parr,” Chemical Physics Letters, Vol. 157, No. 3, 1989, pp. 200-206. doi:10.1016/0009-2614(89)87234-3 |

[103] | R. C. Morrison, “The Non-N-Representability of the Colle-Salvetti Second-Order Reduced Density Matrix,” International Journal of Quantum Chemistry, Vol. 46, No. 4, 1993, pp. 583-587. doi:10.1002/qua.560460406 |

[104] | S. Caratzoulas, “Gaussian Resummation Approximation of the Reference Spin-Reduced Second-Order Density Matrix in the Colle-Salvetti Model for Electron Correlation,” Physical Review A, Vol. 63, No. 6, 2001, Article ID: 062506. |

[105] | S. Ragot, “Assessment of an Analytical Density Matrix Derived from a Modified Colle-Salvetti Approach to the Electron Gas,” Journal of Chemical Physics, Vol. 132, No. 6, 2010, Article ID: 064104. |

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